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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Sard's theorem ： ウィキペディア英語版 Sard's theorem Sard's theorem, also known as Sard's lemma or the Morse–Sard theorem, is a result in mathematical analysis which asserts that the set of critical values (that is, the image of the set of critical points) of a smooth function ''f'' from one Euclidean space or manifold to another is a null set, i.e., it has Lebesgue measure 0. This makes the set of critical values "small" in the sense of a generic property. The theorem is named for Anthony Morse and Arthur Sard. == Statement == More explicitly (; ), let :$f\backslash colon\; \backslash mathbb^n\; \backslash rightarrow\; \backslash mathbb^m$ be $C^k$, (that is, $k$ times continuously differentiable), where $k\backslash geq\; \backslash max\backslash$. Let $X$ denote the ''critical set'' of $f,$ which is the set of points $x\backslash in\; \backslash mathbb^n$ at which the Jacobian matrix of $f$ has rank . Then the image $f(X)$ has Lebesgue measure 0 in $\backslash mathbb^m$. Intuitively speaking, this means that although $X$ may be large, its image must be small in the sense of Lebesgue measure: while $f$ may have many critical ''points'' in the domain $\backslash mathbb^n$, it must have few critical ''values'' in the image $\backslash mathbb^m$. More generally, the result also holds for mappings between second countable differentiable manifolds $M$ and $N$ of dimensions $m$ and $n$, respectively. The critical set $X$ of a $C^k$ function :$f:N\backslash rightarrow\; M$ consists of those points at which the differential :$df:TN\backslash rightarrow\; TM$ has rank less than $m$ as a linear transformation. If $k\backslash geq\; \backslash max\backslash$, then Sard's theorem asserts that the image of $X$ has measure zero as a subset of $M$. This formulation of the result follows from the version for Euclidean spaces by taking a countable set of coordinate patches. The conclusion of the theorem is a local statement, since a countable union of sets of measure zero is a set of measure zero, and the property of a subset of a coordinate patch having zero measure is invariant under diffeomorphism. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Sard's theorem」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.